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Character sheaves for symmetric pairs: spin groups

Ting Xue

TL;DR

This work classifies character sheaves for symmetric pairs attached to spin groups, establishing that all such sheaves arise from the nearby cycle construction and parabolic induction. It provides explicit descriptions of supports and equivariant local systems via braided- and Hecke-algebra representations, and identifies cuspidal as well as nilpotent-support families, with a complete enumeration by central character. The analysis combines detailed orbit theory for type BDI and DIII, extended braid group representations, and generalized nearby cycles to deliver a full classification. The results extend Lusztig-style frameworks to spin-group symmetric pairs, enabling precise computation of character sheaves and their central-character decompositions, with potential applications in geometric representation theory and modular categorical structures.

Abstract

We determine character sheaves for symmetric pairs associated to spin groups. In particular, we determine the cupsidal character sheaves and show that they can be obtained via the nearby cycle construction of [GVX] and its generalisation in [VX2].

Character sheaves for symmetric pairs: spin groups

TL;DR

This work classifies character sheaves for symmetric pairs attached to spin groups, establishing that all such sheaves arise from the nearby cycle construction and parabolic induction. It provides explicit descriptions of supports and equivariant local systems via braided- and Hecke-algebra representations, and identifies cuspidal as well as nilpotent-support families, with a complete enumeration by central character. The analysis combines detailed orbit theory for type BDI and DIII, extended braid group representations, and generalized nearby cycles to deliver a full classification. The results extend Lusztig-style frameworks to spin-group symmetric pairs, enabling precise computation of character sheaves and their central-character decompositions, with potential applications in geometric representation theory and modular categorical structures.

Abstract

We determine character sheaves for symmetric pairs associated to spin groups. In particular, we determine the cupsidal character sheaves and show that they can be obtained via the nearby cycle construction of [GVX] and its generalisation in [VX2].

Paper Structure

This paper contains 23 sections, 17 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Nilpotent orbits and component groups
  4. Type BDI
  5. Type DIII
  6. Representations of extended braid groups
  7. Notations
  8. Nearby cycle sheaves
  9. Nearby cycles for type DIII and split type BDI
  10. Type DIII
  11. Split type BDI
  12. Generalised nearby cycles
  13. Character sheaves
  14. Supports of character sheaves and equivariant fundamental groups
  15. Type BDI
  16. ...and 8 more sections

Key Result

Theorem 1.1

The cuspidal character sheaves are

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • ...and 20 more